Poroelasticity of a covalently crosslinked alginate ...

Poroelasticity of a covalently crosslinked alginate hydrogel undercompressionSengqiang Cai, Yuhang Hu, Xuanhe Zhao, and Zhigang Suo Citation: J. Appl. Phys. 108, 113514 (2010); doi: 10.1063/1.3517146 View online: http://dx.doi.org/10.1063/1.3517146 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v108/i11 Published by the American Institute of Physics. Related ArticlesLiquid pearls Phys. Fluids 23, 091108 (2011) Non-positional cell microarray prepared by shape-coded polymeric microboards: A new microarray format formultiplex and high throughput cell-based assays Biomicrofluidics 5, 032001 (2011) The vibrational density of states of a disordered gel model J. Chem. Phys. 135, 104502 (2011) Effect of surface composition and roughness on the apparent surface free energy of silica aerogel materials Appl. Phys. Lett. 99, 104104 (2011) Selectively solvated triblock copolymer networks under biaxial strain Appl. Phys. Lett. 99, 101908 (2011) Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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Poroelasticity of a covalently crosslinked alginate hydrogelunder compression

Sengqiang Cai,1 Yuhang Hu,1 Xuanhe Zhao,2 and Zhigang Suo1,a�

1School of Engineering and Applied Sciences, Kavli Institute for Nanobio Science and Technology,Harvard University, Cambridge, Massachusetts 02138, USA2Department of Mechanical Engineering and Materials Science, Duke University, Durham,North Carolina 27708, USA

�Received 15 September 2010; accepted 18 October 2010; published online 7 December 2010�

This paper studies the poroelastic behavior of an alginate hydrogel by a combination of theory andexperiment. The gel—covalently crosslinked, submerged in water, and fully swollen—is suddenlycompressed between two parallel plates. The gap between the plates is held constant subsequently,and the force on the plate relaxes while water in the gel migrates. This experiment is analyzed byusing the theory of linear poroelasticity. A comparison of the relaxation curve recorded in theexperiment and that derived from the theory determines the elastic constants and the permeability ofthe gel. The material constants so determined agree well with those determined by using a recentlydeveloped indentation method. Furthermore, during relaxation, the concentration of water in the gelis inhomogeneous, resulting in tensile hoop stresses near the edge of the gel, and possibly causingthe gel to fracture. © 2010 American Institute of Physics. �doi:10.1063/1.3517146�

I. INTRODUCTION

A flexible, covalent network of polymers can imbibe alarge quantity of a solvent, resulting in a gel. Gels constitutemany tissues of animals and plants, and are used in diverseapplications, including drug delivery,1,2 microfluidics,3,4 tis-sue engineering,5,6 oilfield management,7,8 and foodprocessing.9,10 The mechanical behavior of gels11–13 and gel-like tissues �e.g., cartilage�14,15 is time-dependent. The net-work enables large and reversible deformation, while the sol-vent in the gel migrates. The concurrent deformation of thenetwork and migration of the solvent is known as poroelas-ticity.

We have recently reported experiments on an alginatehydrogel pressed by a flat plate16 and by an indenter.17 Ineach experiment, a disk of an alginate hydrogel is covalentlycrosslinked, submerged in water or aqueous solution, andfully swollen. The gel is pressed by suddenly pressing theplate �Fig. 1�a�� or the indenter �Fig. 1�b��. The displacementis kept constant subsequently �Fig. 1�c��, while the force onthe plate or the indenter is recorded as a function of time�Fig. 1�d��. The force instantly rises and then relaxes, aswater in the gel migrates and the gel approaches a new stateof equilibrium. This relaxation curve is used to deduce ma-terial constants of the gel—the shear modulus and Poisson’sratio of the gel, as well as the permeability of the solventthrough the network.

The main object of this paper is to ascertain that the twomethods—compression and indentation—yield the same ma-terial constants for the same gel. To minimize the variabilityof the gel used in the two experiments, here we conduct bothexperiments by using the alginate hydrogel prepared in thesame batch. The material constants of the gel are determinedby comparing the relaxation curves obtained from the experi-

ments to those derived from the theory of poroelasticity. Ourprevious paper17 has reported the theoretical relaxation curvefor indentation, and this paper will derive the theoretical re-laxation curve for compression. Furthermore, we will de-scribe the theoretical prediction of transient fields in the

a�Electronic mail: [email protected].

F(∞)

F

t

F(0)

(c) (d)

t

gel

solvent

(a)

bz

F

F

Impermeable, frictionless

a

F

solvent

indenter

gelmigrationof solvent2a

(b)

FIG. 1. �Color online� �a� A disk of a gel is submerged in a solvent, and iscompressed by frictionless, impermeable, rigid plates. �b� A disk of a gel issubmerged in a solvent, and a conical indenter is pressed into the gel. �c� Inboth experiments, the displacement is suddenly prescribed and subsequentlyheld fixed. �d� The force is recorded as a function of time.

JOURNAL OF APPLIED PHYSICS 108, 113514 �2010�

0021-8979/2010/108�11�/113514/8/$30.00 © 2010 American Institute of Physics108, 113514-1

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compressed gel. In particular, the transient hoop stress istensile near the edge of the gel, and may cause the gel tofracture.

II. GOVERNING EQUATIONS OF POROELASTICITY

This section writes Biot’s theory of poroelasticity18 in aform suitable for the analysis of the compression test. Thepresentation will be brief; details concerning application ofthe theory to polymer gels may be found elsewhere �e.g.,Refs. 12, 13, 19, and 20�. Figure 1 illustrates a disk of a gel,radius a and thickness b, along with the cylindrical coordi-nates �r ,� ,z�. The disk is pressed vertically, and the gel isslippery between the two plates, so that the disk is taken todeform under the condition of generalized plane strain. Theaxial strain is homogeneous in the gel but can vary withtime. Let �z�t� be the axial strain of the gel as a function oftime. The deformation of the disk is taken to be axisymmet-ric, so that the radial displacement u is independent of z and� but is a function of time and radial position. Write the fieldof the radial displacement as u�r , t�. The hoop strain and theradial strain are

�� = u/r , �1�

�r = �u/�r . �2�

All the shear strains vanish.The plates are impermeable to the solvent, and the sol-

vent in the gel migrates in the radial direction. Let J�r , t� bethe flux of the solvent �i.e., the number of solvent moleculescrossing unit area in a reference state per unit time�. LetC�r , t� be the field of the concentration �i.e., the number ofsolvent molecules per unit volume of the gel in the referencestate�. The number of solvent molecules is conserved:

�C

�t+

��rJ�r � r

= 0. �3�

The gel is in mechanical equilibrium at all time. Theradial stress �r�r , t� and the hoop stress ���r , t� satisfy

��r

�r+

�r − ��

r= 0. �4�

The axial stress �z�r , t� gives rise to the compressive force:

F�t� = − 2��0

a

�zrdr . �5�

We adopt the sign convention that the compressive force F ispositive. All components of the shear stresses vanish.

The gel, however, is not in diffusive equilibrium. Thechemical potential of the solvent in the gel is a time-dependent field ��r , t�. The gradient of the chemical poten-tial �� /�r drives the flux of the solvent. The two quantitiesare taken to be linearly related, written in the form

J = −k

��2

��

�r, �6�

where � is the viscosity of the solvent and � the volume persolvent molecule. Both � and � are taken to be the values

for the pure liquid solvent �e.g., for water �=1.010−3 N s m−2 and �=3.010−29 m3�. Consequently, �6�defines a phenomenological quantity, k, which is known asthe permeability and has the dimension of length squared.

At any time, each differential element of the gel is in astate of thermodynamic equilibrium. A reference state is as-signed when the gel is stress-free and the solvent in the gel isin equilibrium with the pure liquid solvent. In the referencestate, the strains of the gel are set to be zero, the chemicalpotential of the solvent in the gel is set to be zero, and theconcentration of the solvent in the gel is denoted by C0.When the gel is subject to a state of stress, the gel is inanother state of equilibrium, in which the gel deforms andthe solvent in the gel may no longer be in equilibrium withthe pure liquid solvent. This state of equilibrium of the gel ischaracterized by the stresses ��r ,�� ,�z�, the strains��r ,�� ,�z�, the concentration C, and the chemical potential ofthe solvent �. These thermodynamic variables are connectedthrough the equations of state, as described below.

Because the stress in a gel is typically small, the poly-mers and the solvent molecules are commonly assumed to beincompressible. Consequently, the increase in the volume ofthe gel is entirely due to the volume of the absorbed solvent:

�r + �� + �z = ��C − C0� . �7�

The gel is assumed to be isotropic, and the stresses are as-sumed to be linear in strains. Under these assumptions, theequations of state take the form17

�r = 2G��r +

1 − 2��r + �� + �z�� −

�

�, �8�

�� = 2G��� +

1 − 2��r + �� + �z�� −

�

�, �9�

�z = 2G��z +

1 − 2��r + �� + �z�� −

�

�, �10�

where G is the shear modulus and Poisson’s ratio. Whenthe gel is constrained by rigid and permeable walls in alldirections, such that all strains vanish, an increase in thechemical potential of the solvent gives rise to a hydrostaticpressure, � /�.

The above equations specify the theory of poroelasticity.A combination of these equations gives the governing equa-tions for the fields C�r , t�, u�r , t�, and ��r , t�:

��ru�r � r

+ �z�t� = ��C − C0� , �11�

2�1 − ��1 − 2�

�

�r� ��ru�

r � r� =

��

G� � r, �12�

�C

�t=

D�

r � r r � C

�r , �13�

with the diffusivity given by

113514-2 Cai et al. J. Appl. Phys. 108, 113514 �2010�

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D =2�1 − �Gk

�1 − 2��. �14�

Equation �13� takes the familiar form of the diffusion equa-tion. In poroelasticity, however, this diffusion equation can-not be solved by itself, because the boundary conditions typi-cally involve the chemical potential and the displacement.Nonetheless, �13� indicates that over time t a disturbancediffuses over a length �Dt.

Throughout the experiment, the gel is submerged in apure liquid solvent, whose chemical potential is set to bezero. Before being compressed, the gel is in equilibrium withthe external solvent—a state taken to be the reference state ofthe gel. At time t=0, a compressive strain of magnitude � issuddenly prescribed by pressing the rigid plates, and thisstrain is held constant in subsequent time. That is, �z�t�=−�,for t�0. We adopt the sign convention that ��0 for com-pression.

The boundary conditions on the edge of the disk areobtained by assuming that the gel is locally in equilibriumwith the external solvent at all time. Thus, the chemical po-tential of the solvent in the gel, on the edge, equals that ofthe external solvent at all time:

��a,t� = 0. �15�

Furthermore, the radial stress on the edge of the gel vanishesat all time:

�r�a,t� = 0. �16�

Inserting �15� and �16� into �8�, we obtain a boundary con-dition in terms of the displacement:

�1 − ��u

�r�a,t� + �u�a,t�

r− �� = 0. �17�

III. SHORT-TIME AND LONG-TIME LIMITS

The compression causes a portion of the solvent in thegel to migrate out, so that the field in the gel evolves withtime. We first consider the short-time limit, t=0, instanta-neously after the gel is compressed with the strain �. The gelundergoes a homogeneous deformation. Instantaneously afterthe gel is compressed, the solvent in the gel has no time tomigrate so that C�r ,0�=C0, and the volume of the gel doesnot change, �r+��+�z=0. The axial strain is �z=−�, and theradial and hoop strains are

�r�r,0� = ���r,0� =�

2. �18�

The radial displacement is

u�r,0� =1

2�r . �19�

Instantaneously after compression, the radial and the hoopstresses are zero, �r�r ,0�=���r ,0�=0. The solvent in the gelis out of equilibrium with the external solvent: the chemicalpotential of the solvent in the gel is homogeneous but is notzero. Setting �r�r ,0�=0 and �r+��+�z=0 in Eq. �8�, we ob-tain that

��r,0� = �G� . �20�

From �10� we obtain the axial stress

�z�r,0� = − 3G� . �21�

Recall that the edge of the gel is assumed to be in localequilibrium with the external solvent at all time, so that��a ,0�=0 instantaneously after compression. This boundaryvalue is unequal to the value in the interior of the gel,��r ,0�=�G�. Such a discontinuity is common in initial/boundary-value problems subject to suddenly prescribed ini-tial conditions. We now examine the consequence of thisdiscontinuity in the chemical potential. Geometric compat-ibility requires that ���a ,0�=� /2, while mechanical equilib-rium requires that �r�a ,0�=0. Inserting these conditions,along with ��a ,0�=0, into �8�–�10�, we obtain the instanta-neous radial strain

�r�a,0� =�

2�1 − �, �22�

the hoop stress

���a,0� =1 − 2

1 − G� , �23�

and the axial stress

�z�a,0� = −2 −

1 − G� . �24�

The radial strain on the edge �22� differs from that in theinterior of the gel, �r�r ,0�=� /2. Similarly, the hoop andaxial stresses also differ from their counterparts in the inte-rior of the gel. Also note that the instantaneous hoop stress�23� on the edge of the gel is tensile.

We next consider the long-time limit, t→ . After beingcompressed for a long time, the gel reaches a new state ofequilibrium: the chemical potential of the solvent every-where in the gel equals that in the external solvent, ��r , �=0. The radial and hoop stresses vanish, �r�r , �=���r , �=0. From Eqs. �8� and �9�, we obtain the radial and the hoopstrains:

�r�r, � = ���r, � = � . �25�

The radial displacement is

u�r, � = �r . �26�

Equation �10� gives the axial stress

�z�r, � = − 2�1 + �G� . �27�

A comparison of �18� and �25� shows that, as the solventmigrates out the gel, the transverse expansion reduces fromthe instantaneous value �r�r ,0�=���r ,0�=� /2, and ap-proaches the value of a new state of equilibrium, �r�r , �=���r , �=�. Thus, Poisson’s ratio characterizes the ch-emomechanical interaction of the gel. Poisson’s ratio is re-stricted in the interval −1��1 /2 by the requirement thatthe free-energy density is positive definite. When a gel issubject to compression and reaches a new state of equilib-

113514-3 Cai et al. J. Appl. Phys. 108, 113514 �2010�

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rium with the external solvent, no solvent in the gel migratesout if →1 /2, or portion of the solvent in the gel migratesout if −1��1 /2.

IV. TRANSIENT FIELDS

The partial differential Eqs. �11�–�13�, along with theboundary conditions �15� and �16� and the initial conditions�19� and �20�, can be solved by the method of separation ofvariables. Consider displacement fields of,

u�r,t� = �r + f�r�exp�− �t� . �28�

The first term is the long-time limit, and the second termrepresents the transient deviation from the new state of equi-librium. Inserting �28� into �11� and �13�, we obtain that

Dd

dr�d�rf�

rdr� + �f = 0, 0 � r � a . �29�

The solution to this ordinary differential equation is theBessel functions. Let Jm��� be the Bessel function of orderm. The displacement field that solves �11�–�13�, �15�, �16�,�19�, and �20� is

u�r,t�a�

= r

a + �

n=1

BnJ1�nr

aexp− �n

2Dt

a2 . �30�

The eigenvalues �n are determined by �17�, namely,

�1 − ��nJ1���n� + J1��n� = 0. �31�

The coefficients Bn are determined by the initial condition�19�, giving

Bn =�1 − 2��1 − �2�n

�n2�1 − �2 − �1 − 2�

J2��n�J1

2��n�. �32�

Figure 2 plots the displacement field at several times. Instan-taneously after the gel is compressed, at Dt /a2=0, the diskexpands in the radial direction, and the displacement is linearin the radius, u�r ,0�=�r /2. As time proceeds, solvent gradu-ally migrates out the gel, and the disk shrinks. When Dt /a2

→ , the gel attains the new state of equilibrium, u�r ,0�=�r. Figure 2 indicates that the gel nearly attains the newstate of equilibrium when Dt /a2=1.

Inserting the displacement field �30� into �12�, and usingthe boundary condition ��a , t�=0, we obtain the field ofchemical potential:

��r,t�G��

=2�1 − �1 − 2

�n=1

Bn�n�J0�nr

a − J0��n��

exp− �n2Dt

a2 . �33�

Figure 3 plots the chemical potential field at several times.Immediately after compression, the chemical potential of thesolvent inside the gel is homogenous, ��r ,0�=�G�. Thischemical potential exceeds the chemical potential of solventoutside the gel, �=0, and drives the solvent to migrate out.The chemical potential of the solvent in the gel at the edge ofthe disk is taken to equal that in the external solvent at alltime, ��a , t�=0. As time proceeds, the chemical potential ofthe solvent in the gel gradually decreases. In the long-timelimit, the compressed gel equilibrates with the external sol-vent, and the chemical potential of the solvent in the gelapproaches zero.

Inserting the displacement field �30� and the chemicalpotential field �33� into the equations of state �8�–�10�, weobtain the stresses:

��

G�= �

n=12Bn a

rJ1�n

r

a + �n�− J0�n

r

a

+1 −

1 − 2J0��n���exp− �n

2Dt

a2 , �34�

�r

G�= �

n=12Bn�−

a

rJ1�n

r

a +

1 −

1 − 2�nJ0��n��

exp− �n2Dt

a2 , �35�

�z

G�= − 2� + 1� + �

n=12Bn�n�− J0�n

r

a

+1 −

1 − 2J0��n��exp− �n

2Dt

a2 . �36�

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

�au

ar /

02

�a

Dt

2.02

�a

Dt

4.02

�a

Dt

12

�a

Dt

3

1��

� � rru �3

1, ��

FIG. 2. �Color online� The distribution of the radial displacement at severaltimes.

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

ar /

εμΩG 0

2=

a

Dt

2.02=

a

Dt

4.02=

a

Dt

6.02=

a

Dt

0.12=

a

Dt8.0

2=

a

Dt

FIG. 3. �Color online� The distribution of the chemical potential of thesolvent in the disk at several times.

113514-4 Cai et al. J. Appl. Phys. 108, 113514 �2010�

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Figure 4�a� plots the distribution of the hoop stress atseveral times. Instantaneously after compression, the solventin the gel has no time to migrate, so that no hoop stressdevelops in the interior of the disk. However, a tensile hoopstress develops instantaneously at the edge of the disk, asdiscussed before. After the gel is compressed for some time,solvent leaves the gel gradually, so that the concentrationbecomes inhomogeneous: the concentration of the solventnear the edge is lower than that around the center. As timeproceeds, the tensile stress reduces magnitude but spreadsover a larger region. Toward the center of the disk, the hoopstress is compressive. In the long-time limit, the hoop stresseverywhere in the gel vanishes.

Figure 4�b� plots the distribution of the radial stress atseveral times. The radial stress at the edge vanishes at alltime, as dictated by the boundary condition. After the gel iscompressed by the plates, the solvent migrates out, initiallyfrom the region near the edge of the disk. Consequently, theradial stress around the center of the disk is compressive.

The magnitude of the compressive radial stress initially risesand then falls. In the long-time limit, the radial stress every-where vanishes.

Figure 4�c� plots the distribution of the axial stress atseveral times. As discussed before, instantaneously after thegel is pressed, the axial stress is �z�a ,0�=−G��2−� / �1−� at the edge of the disk, and is �z�r ,0�=−3G� in theinterior of the disk. These two levels of the axial stress areunequal, so long as �0.5. The magnitude of the axial stressat the edge of the disk increases as time progresses. Themagnitude of the axial stress at the center of the disk initiallyrises and then falls. After some time, the axial stress homog-enizes in the disk, and approaches the long-time limit�z�r , �=−2�1+�G�.

V. USING RELAXATION CURVES TO DETERMINEPROPERTIES OF GELS

Integrating the axial stress over the area of the disk, weobtain the axial force as a function of time:

F�t�G��a2 = 2� + 1� − �

n=1

Bn− 4J1��n�

+2�1 − �1 − 2

�nJ0��n�exp− �n2Dt

a2 . �37�

The short-time limit is

F�0� = 3�G�a2. �38�

The long-time limit is

F� � = 2��1 + �G�a2. �39�

Figure 5 plots the relaxation curve �37� in the form

F�t� − F� �F�0� − F� �

= f,Dt

a2 , �40�

The ratio on the left-hand side measures how far the gel isaway from the state of equilibrium. The ratio depends onPoison’s ratio weakly, as indicated in Fig. 5.

Covalently crosslinked alginate hydrogels are preparedfollowing the protocol previously described.21 The gel is sub-

0 0.2 0.4 0.6 0.8 1-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

3

1��

05.02�

a

Dt

1.02�

a

Dt

0.12�

a

Dt

6.02�

a

Dt

4.02�

a

Dt

2.02�

a

Dt

ar /

r�

r�

(b)

0 0.2 0.4 0.6 0.8 1

-0.1

0

0.1

0.2

0.3

0.4

0.5

05.02�

a

Dt

1.02�

a

Dt

2.02�

a

Dt

0.12�

a

Dt

6.02�

a

Dt

4.02�

a

Dt

)0,(a��

3

1��

(a)

gel disk

��

��

ar/

ar /0 0.2 0.4 0.6 0.8 1

-3.2

-3.1

-3

-2.9

-2.8

-2.7

-2.6

-2.5

-2.4

02�

a

Dt

6.02�

a

Dt

05.02�

a

Dt1.0

2�

a

Dt

4.02�

a

Dt

2.02�

a

Dt

0.12�

a

Dt

)0,(az�3

1��

(c)

FIG. 4. �Color online� The evolution of �a� the hoop stress, �b� the radialstress, and �c� the axial stress.

FIG. 5. �Color online� The compressive force relaxes as a function of time.The relaxation curve varies with Poisson’s ratio somewhat.

113514-5 Cai et al. J. Appl. Phys. 108, 113514 �2010�

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merged in distilled water for 24 h until it is fully swollen.From a piece of the gel, we punch out three disks of radii 3,4, and 5 mm. These disks are then pressed with a stainlesssteel plate. The plate first approaches the surface of the diskwith a speed of 2 �m /s until the measured force starts toincrease. At this point, the gap between the top and bottomplates is viewed as the thickness of the disk, which is 7.71,7.82, 7.65 mm, of 3 mm radius, 4 mm radius, and 5 mmradius sample, respectively. Each disk is subject to a 20%vertical compressive strain. The total rising time is about 10s, which is negligible compared to the relaxation time �about3 to 8 h�. While the vertical strain is held at the fixed value,the force on the plate is recorded as a function of time byusing the AR rheometer from TA Instruments. The resolutionof the force is 0.005 N, and data are taken at the rate of 360points per second.

The covalently crosslinked alginate hydrogels are quitebrittle, and sometimes fracture during the experiment �Fig.6�. The fracture mechanics of gels is interesting in its ownright but will not be pursued in this paper. The data reportedbelow are taken from experiments in which no fracture isobserved.

Figure 7�a� plots the relaxation curves measured experi-mentally from the three disks. In each case, the force risessharply as the plate is pressed. Subsequently, the plate is heldat the fixed position, while the force relaxes and approachesa new state of equilibrium. The magnitude of the force, aswell as the relaxation time, is larger when the radius of thedisk is larger. Once the force is divided by the area of thedisk �a2, and the time is divided by a2, the relaxation curvesmeasured from the disks of the three radii collapse into asingle curve �Fig. 7�b��. This behavior is consistent with theprediction of the theory of poroelasticity. The nominalstress—the force divided by the area of the disk—relaxes asthe solvent migrates out from the edge of the gel. The relax-ation time is proportional to the radius of the disk squared.

By comparing the relaxation curve measured experimen-tally with that derived from the theory of poroelasticity, wecan determine the shear modulus, Poisson’s ratio, and thediffusivity. In the short-time limit, a comparison of the ex-perimental data F�0� /�a2=20.5 kPa and the theoretical for-mula F�0� /�a2=3G� gives the shear modulus G=34.2 kPa. In the long-time limit, a comparison of the ex-perimental data F� � /F�0�=0.82 and the theoretical formulaF� � /F�0�=2�1+� /3 gives Poisson’s ratio =0.23. The re-laxation curve calculated from the theory of poroelasticityoverlaps with the relaxation curves experimentally measuredfrom the three disks when the diffusivity is fit to D=6.210−9 m2 /s �Fig. 7�b��.

In a recent paper,17 we have used a conical indenter tocharacterize the alginate hydrogel. As illustrated in Fig. 1,the gel is submerged in water and is fully swollen. The coni-cal indenter, of half included angle �, is suddenly pressedinto the gel and is subsequently held at a fixed depth h. Theforce on the indenter is measured as a function of time. Thistest has been analyzed within the theory of poroelasticity,17

and the relevant results are summarized here. The radius ofcontact is given by

a =2

�h tan � . �41�

In the short-time limit, solvent in the gel has no time tomigrate, the gel behaves like an incompressible elastic solid,and the force on the indenter is given by

F�0� = 4Gah . �42�

In the long-time limit, portion of the solvent in the gel hasmigrated out, the gel has attained a new state of equilibriumwith the external solvent, and the force on the indenter isgiven by

F� � = 2Gah/�1 − � . �43�

For the gel to evolve from the short-time limit toward thelong-time limit, the solvent in the gel under the indenter mustmigrate. The relevant length in this diffusion-type problem is

FIG. 6. �Color online� Photos of fractured alginate hydrogel caused bycompression.

FIG. 7. �Color online� A disk of an alginate hydrogel is compressed betweenparallel plates, while the force on the planes is recorded as a function oftime. �a� Relaxation curves obtained by using disks of an alginate hydrogelof three radii. �b� Each of the three relaxations curves is plotted again, withthe force divided by the area of the disk, and the time divided by the radiussquared. Also plotted is the relaxation curve obtained from the theory ofporoelasticity.

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the radius of contact, a, and the normalized time takes theform �=Dt /a2. The function F�t� obeys

F�t� − F� �F�0� − F� �

= g��� . �44�

The dimensionless ratio on the left-hand side of �44� is ameasure of how far the gel is away from the new state ofequilibrium. The function g��� is determined by solving theporoelastic boundary-value problem. Our previous work in-dicates that g is a function of the single variable �, given by

g��� = 0.493 exp�− 0.822��� + 0.507 exp�− 1.348�� .

�45�

To minimize the variability of the material, we make thealginate hydrogels for both tests—compression andindentation—in the same batch. The solutions are pouredinto a plastic mold of 3 cm radius and 2 cm thickness. Aftergelation, the gel is submerged in distilled water for 48 h untilit is fully swollen. We then press an aluminum indenter ofhalf included angle �=70° into the gel to a fixed depth. Theforce on the indenter is recorded as a function of time byusing a custom-built load frame with a force resolution of0.01 N and a displacement resolution of 1 �m. The indent-ers are programmed to approach the surface of the sample atthe speed of 2 �m /s, until the slope of the recorded force-displacement curve start to be positive. The time used topress the indenter into the alginate gels �10 s� is much shorterthan the relaxation time �3 to 16 h�, so that the effect of theinitial loading stage is minimized.

Figure 8�a� shows the measured relaxation curves re-

corded at the three depths of indentation. In each case, theforce rises sharply, and then relaxes as the gel approaches anew state of equilibrium with the external solvent. The mag-nitude of the force, as well as the relaxation time, is largerwhen the depth of indentation is larger. A comparison of theexperimental value F�0� /ah=130 kPa and the analytical for-mula F�0� /ah=4G gives G=32.5 kPa. A comparison of theexperimental value F�0� /F� �=1.56 and the analytical for-mula F�0� /F� �=2�1−� gives Poisson’s ratio =0.22.

Figure 8�b� plots the relaxation curves measured with thethree depths indentation the in a dimensionless form. Thethree curves collapse into a single curve. Furthermore, thesecurves overlap with the relaxation curve calculated with thetheory of poroelasticity, g��� in �45�, when the diffusivity isfit to the value D=6.610−9 m2 /s.

Comparing the material properties measured by com-pression and indentation, we note 5.2% difference in theshear modulus, 4.6% difference in Poisson’s ration, and 6%in the diffusivity. This excellent agreement lends support toboth tests. The two tests have their own advantages and dis-advantages. The compression test requires the sample to befabricated with perfectly parallel top and bottom surfaces,which may be difficult in practice. This concern is absent forthe indentation test because the starting point of the measure-ment is readily detected for conical or spherical indenters.The indentation test, however, requires the thickness of thesample to be more than ten times larger than the depth ofindentation. This requirement might be difficult to satisfy inpractice.

VI. CONCLUDING REMARKS

The compression test is analyzed within the theory ofporoelasticity. By comparing the relaxation curve derivedfrom the theory to that measured in the experiment, we ob-tain the shear modulus, Poisson’s ratio and the permittivityof a covalently crosslinked alginate hydrogel. The materialconstants so determined agree well with those obtained froma recently developed indentation method. The agreementlends support to both methods. Furthermore, our calculationshows that, as the compressed gel relaxes, the concentrationof the solvent in the gel is inhomogeneous, resulting in ten-sile hoop stresses near the edge of the gel. While fracture isindeed often observed in our experiments, the mechanics offracture awaits clarification.

ACKNOWLEDGMENTS

This work is supported by the NSF �Grant No. CMMI-0800161� and by the MRSEC at Harvard University.

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FIG. 8. �Color online� A conical indenter is pressed into a disk of an algi-nate hydrogel to a certain depth, while the force on the indenter is recordedas a function of time. �a� Relaxation curves obtained by keeping the indenterat three depths. �b� The relaxation curves are plotted again by using normal-ized variables. Also included is the relaxation curve obtained from thetheory of poroelasticity.

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